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A traffic signal board, indicating ‘SCHOOL AHEAD’, is an equilateral triangle with side ‘*a*’. Find the area of the signal board, using Heron’s formula. If its perimeter is 180 cm, what will be the area of the signal board?

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#### Solution

Side of traffic signal board = *a*

Perimeter of traffic signal board = 3 × *a*

`2s = 3a rArr s = 3/2 a`

By heron's formula,

`"Area of triangle "=sqrt(s(s-a)(s-b)(s-c))`

`"Area of given triangle "= sqrt(3/2 a(3/2a-a)(3/2a-a)(3/2a-a))`

`= sqrt((3/2a)(a/2)(a/2)(a/2))`

`= sqrt3/4a^2" ..........(1)"`

Perimeter of traffic signal board = 180 cm

Side of traffic signal board **(a)=(180/3)cm = 60 cm**

Using equation (1), area of traffic signal board

`= sqrt3/4(60cm)^2`

`= (3600/4sqrt3)cm^2 = 900sqrt3 cm^2`

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